Finsler Geometry, Graph Neural Networks, and You
T. Mitchell Roddenberry, Richard G. Baraniuk
cs.LG, eess.SP, math.DG, stat.ML
Submitted: 2026-06-15
Code: https://github.com/ax-ml/jax
License: http://creativecommons.org/licenses/by/4.0/
The gist: Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators.
Terminology
Abstract
Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.
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